Simply connectedness of Kähler and Riemannian manifolds via spectral estimates (with an appendix by Shiyu Zhang)

Let $(M,h)$ be a compact Kähler manifold. Under a rather weak spectral positivity assumption we prove that $M$ is rationally connected and thus simply connected, projective with $h^{p,0}(M)=\{0\}$ for each $p>0$. Then, in the second part of this paper, we focus on Riemannian manifolds and we provide an appropriate spectral positivity assumption which guarantees that a compact and oriented even dimensional Riemannian manifold $(M,g)$ is a simply connected real homology sphere. Finally, in the appendix, a characterization of the rational dimension of compact Kähler manifolds in terms of the positivity of the minimal slope of the tangent bundle is given.